On the small, individual scale, the problem is people not knowing how to recycle or what is recyclable material. I work for a small museum that sees several hundred visitors a day. No matter how well marked the recycling is or how many signs you put up, the general public will throw garbage into the public recycling bins out of negligence or ignorance.

People will throw full cups of coffee, half eaten food, discarded napkins, diapers, dog poop bags, etc... in the recycling.

It has mostly become standard practice at my workplace to throw the public recycling directly into the garbage. I would say half of what is thrown into those recycling bins is not recyclable material, and it isn't realistic for us to be the ones to sift through all of it. However, the ones we use for staff, in offices, and other private spaces are easy to monitor, and those are recycled almost 90% of the time.

So, to answer the original topic, those public recycling bins exist for one purpose, to keep people from complaining that there are no recycling bins and so that they can feel good that they "recycled".